This ebook comprises lecture notes of a summer time university named after the past due Jacques Louis Lions. The summer time tuition was once designed to alert either Academia and to the expanding function of multidisciplinary tools and instruments for the layout of advanced items in a variety of components of socio-economic curiosity.

Theorem (The monotone convergence principle). Let {an } be an increasing sequence which is bounded above; then {an } is a convergent sequence. Let {an } be a decreasing sequence which is bounded below; then {an } is a convergent sequence Proof. 2. Details will be given in third year, or you can look in (Spivak 1967) for an accurate deduction from the appropriate axioms for R . CHAPTER 3. MONOTONE CONVERGENCE 24 This is a very important result. It is the first time we have seen a way of deducing the convergence of a sequence without first knowing what the limit is.

Then g ◦ f is continuous at a Proof. Pick > 0. We must find δ > 0 such that if |x − a| < δ, then g(f (x)) − g(f (a))| < . We find δ using the given properties of f and g. Since g is continuous at f (a), there is some δ1 > 0 such that if |y − f (a)| < δ1 , then |g(y) − g(f (a))| < . Now use the fact that f is continuous at a, so there is some δ > 0 such that if |x − a| < δ, then |f (x) − f (a)| < δ1 . Combining these results gives the required inequality. 18. Example. 8 is continuous everywhere.